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 September 30th, 2015, 11:06 PM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 104 Thanks: 1 height of prime ideal can someone please help me to show this result: Let A be a factoriel ring,and P included in A be a prime ideal show this equivalence: a)p is a principal ideal b) ht(p) ≤1 thanks in advance October 11th, 2015, 11:56 AM #2 Newbie   Joined: Oct 2015 From: Toronto Posts: 14 Thanks: 1 Suppose that $P$ is a principal prime ideal generated by $a$, let $Q$ be a prime strictly contained in $P$ and $b$ an element of $Q$. We can write $b =ca$. Since A is factoriel, $c = p_1...p_n$ where $p_i$ is irredcible. We deduce that $b = p_1...p_na$. Thus, $p_1(p_2...p_na)$ is an element of $Q$. Since $Q$ is a prime, $p_1\in Q$ or $p_2...p_nc \in Q$. Suppose that $p_1\in Q$ since $Q$ is contained in $P$, we can write $p_1 = da$ we deduce that $d$ is invertible since $p_1$ is irreducible.This implies that $a\in Q$ and henceforth $P = Q$.This is a contradiction. Thus $p_2(p_3..p_n)a \in Q$. We repeat the previous step $n-1$ times to deduce that $a\in Q$. This is a contradiction thus the height of $P$ is 1. On the other hand, suppose that the height of the prime ideal $P$ is 1. Let $a$ be a non zero element of $P$, we can write $a = p_1...p_n$ where $p_i$ is irreducible since $A$ is factoriel, suppose that $p_i$ is not in $P$ for every $i$, this implies that $p_1p_2$ is not in $P$ since $P$ is a prime. For the same reason, $p_1p_2p_3,....,p_1...p_n =a$ is not in $P$ a fact which is a contradiction. Thus $P$ contains an irreducible element that we call $a$ Since $(a)$ the ideal generated by $a$ is a prime (since $A$ is factoriel and $a$ irreducible) and the height of $P$ is 1, we deduce that $(a) = P$. Last edited by Aristide Tsemo; October 11th, 2015 at 12:00 PM. Tags height, ideal, prime Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post rayman Abstract Algebra 0 September 24th, 2012 04:34 AM talisman Abstract Algebra 2 August 12th, 2012 12:17 AM ely_en Real Analysis 0 March 4th, 2012 11:22 AM iamiam Abstract Algebra 2 December 5th, 2010 12:48 PM nilap Number Theory 2 April 14th, 2009 04:46 PM

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